SpaceX only did crewed flights to orbit after a huge influx of NASA/Space Act money and could lend against future milestone payments as part of the CRS contract. It also had a lot of handed down tech, like the FASTRAC engine design. It really blurs the boundaries between "actually private" and "aren't truly private projects".

It was, however, first to reach orbit uncrewed on purely private funding (Merlin-1 was still heavily based on donated taxpayer technology) with the last flight of Falcon 1. The CRS big bucks enabled SpaceX to skip Falcon 5 for Falcon 9.

Scaled Composites won the Ansari X-Prize by flying suborbital, which was also a privately funded SpaceShipOne and White Knight. It was the first ever privately funded spaceflight with entirely privately developed technology. The next company to achieve the same was Blue Origin.